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- Publisher
- Springer Journals
- Copyright
- Copyright © Tusi Mathematical Research Group (TMRG) 2021
- ISSN
- 2662-2009
- eISSN
- 2538-225X
- DOI
- 10.1007/s43036-021-00174-x
- Publisher site
- See Article on Publisher Site
Abstract
Let p(z)=zn+an-1zn-1+an-2zn-2+⋯+a1z+a0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\cdots +a_1z+a_0$$\end{document} be a complex polynomial with a0≠0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a_0\ne 0$$\end{document} and n≥3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 3$$\end{document}. Several new upper bounds for the moduli of the zeros of p are developed. In particular, if α=∑j=0n-1|aj|2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha =\sqrt{\sum _{j=0}^{n-1}|a_j|^2}$$\end{document} and z is any zero of p, then we show that |z|2≤cos2πn+1+|an-2|+14|an-1|+α2+12α2-|an-1|2+12α,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} |z|^2 \le \cos ^2 \frac{\pi }{n+1}+|a_{n-2}|+ \frac{1}{4} \left( |a_{n-1}|+ { \alpha } \right) ^2 + \frac{1}{2}\sqrt{\alpha ^2-|a_{n-1}|^2} + \frac{1}{2}{\alpha }, \end{aligned}$$\end{document}which is sharper than the existing bound, given as, |z|2≤cos2πn+1+14|an-1|+α2+α,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} |z|^2\le & {} \cos ^2 \frac{\pi }{n+1}+ \frac{1}{4} \left( |a_{n-1}|+ { \alpha }\right) ^2 + {\alpha }, \end{aligned}$$\end{document}if and only if 2|an-2|<∑j=0n-1|aj|2-∑j=0n-2|aj|2.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2|a_{n-2}|< \sqrt{\sum _{j=0}^{n-1}|a_j|^2}-\sqrt{\sum _{j=0}^{n-2}|a_j|^2}.$$\end{document} The upper bounds obtained here enable us to describe smaller annuli in the complex plane containing all the zeros of p.
Journal
Advances in Operator Theory – Springer Journals
Published: Jan 1, 2022
Keywords: Zeros of polynomials; Frobenius companion matrix; 26C10; 15A60